Problem: Find $ \lim_{x\to -5}h(x)$ for $h(x)=\dfrac{5x+4}{x+8}$.
Solution: $h$ is a rational function. Rational functions are continuous across their entire domain, and their domain is all real $x$ -values that don't make the denominator equal to zero. In other words, for any rational function $r$ and any input $c$ in the domain of $r$, we know that this equality holds: $\lim_{x\to c}r(x)=r(c)$ The input $x=-5$ is within the domain of $h$. Therefore, in order to find $ \lim_{x\to -5}h(x)$, we can simply evaluate $h$ at $x=-5$. $\begin{aligned} &\phantom{=}h(x) \\\\ &=\dfrac{5x+4}{x+8} \\\\ &=\dfrac{5(-5)+4}{(-5)+8} \gray{\text{Substitute }x=-5} \\\\ &=\dfrac{-21}{3} \\\\ &=-7 \end{aligned}$ In conclusion, $ \lim_{x\to -5}h(x)=-7$.